You've asked a subtle question. The planets don't move like gears in a clock so no, they never need to have a "meeting spot". When mechanical items move according to chains or gears, all motions occur at rates which can be expressed as fractions or ratios. For instance, the minute hand of a clock moves at exactly 1/60th the rate of the hour hand, and so forth. With such a clockwork mechanism it's possible to obtain exact alignments of the type in the Hercules movie.
But Nature, or Real Life, is more subtle than a TV show. Orbiting bodies in general do not coincide with one another in ratios that can be expressed as simple ratios. As a result, an exact alignment never precisely occurs. If you wait long enough, though, you can get "near enough" to certain types of alignments. How you define "near enough" determines how long you have to wait in between alignments. A nice review by John Savard in Edmonton shows the different ways planets can line up and makes some calculations coming up with a line-up of all the planets happening about every 18,000 years. Note this is longer than all recorded human history.
If you just take the brightest planets, the ones that can be seen with the naked eye--Mercury, Venus, Mars, Saturn and Jupiter--they kind of lined up recently, in May 2002. At least they were near each other in the same part of the sky, in the constellations of Taurus and Gemini. Check out this NASA webpage with a photo of the planets in allignment. The next time this will happen is in 2060. Note however that they are not very precisely lined up, spanning about 40 degrees of arc.
Because of this, scientists do not find this sort of question very interesting. After all, it simply depends on how far you are willing to relax your definition of "close" as in close to each other, or close to the line. From a scientific viewpoint the planets never line up precisely.
What *is* interesting to scientists is when the rates of motions of orbiting bodies *do* become expressible in terms of common fractions. In such cases, the orbital motions are said to be "in resonance". When this happens, the interactions are generally strong enough to affect each other's orbits. The only orbital resonance between the planets exists between Neptune and Pluto. Pluto orbits the sun twice to Neptune's three go-rounds. Among the moons of the planets, though, there are lots of resonances. This is what causes the gaps in the rings of Saturn, or the way the period of rotation of the moon is the same as the period of its orbit so that it always shows the same face to the Earth. Such resonances are important in judging the long-term stability of the orbits of both the planets and their moons.
Calculation of the movements of moons and planets requires mathematics more complex than simple clockwork ratios. The numbers involved are NOT exact rational number ratios like 1/60 but are in fact based on transcendental numbers like pi which have no "terminating" value that can be written out in an engineering calculation to predict exact alignments. Very small interactions between orbiting planets, so small as to be unknowable, could, in certain circumstances eventually have a profound effect--something scientists call "chaos" (in the mathematical sense, not the chaos of darkness from literature).
The study of interactions between orbits and their associated resonances ultimately comes down to the eternal question of how long the solar system can hold together and whether it is stable. Don't worry. It IS stable over timescales much much longer than our lives, although chaos theory does predict a significant probability that the planet Mercury could be ejected from the solar system in a few billion years.
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